Question about Degrees of Freedom

I generally know the concept of degrees of freedom based on the commonly used explanation about how using the average of a sample reduces the the avaialble choices by 1. For example, I generally understand what is explained here.

What is not clear to me is how having more choices for a value or freedom helps in a statistical procedure, say, the hypothesis testing or the regression.

What is not clear to me is how having more choices for a value or freedom helps in a statistical procedure, say, the hypothesis testing or the regression.

More degrees of freedom doesn't necessarily help - in the sense of making a statistical procedure easier to compute or more reliable. Statistical methods rely on probability distributions. Random variables with different degrees of freedom with respect to sample values usually have different probability distributions. The important thing in a given problem is to use the probability distribution that has the correct degrees of freedom.

If you don't have "a given problem" and are instead designing an experiment and thus inventing the statistical problems to be solved then its an interesting question whether you should create problems with large degrees of freedom or small degrees of freedom. It seems to me that, in general, it's simpler to have problems with small degrees of freedom.